# PMSM Current Reference Generator

Permanent magnet synchronous machine current reference generator

**Library:**Simscape / Electrical / Control / PMSM Control

## Description

The PMSM Current Reference Generator block implements a current
reference generator for permanent magnet synchronous machine (PMSM) current control in
the rotor *d*-*q* reference frame.

You typically use this block in a series of blocks making up a control structure.

You can generate a voltage reference in the d-q frame by placing this block before a PMSM Current Control or PMSM Current Control with Pre-Control block.

You can implement velocity control by placing this block after a Velocity Controller block.

You can see an example of a full control structure, from machine measurements to machine inputs, in the PMSM Field-Oriented Control block.

### Equations

The PMSM Current Reference Generator block can obtain the current reference using one of these methods:

Zero

*d*-axis control (ZDAC)User defined lookup tables

Automatically generated lookup tables

For the ZDAC method, the block sets the *d*-axis current
reference ${i}_{d}^{ref}$ to zero and determines the *q*-axis current
reference ${i}_{q}^{ref}$ using the torque equation:

${i}_{d}^{ref}=0,$

and

${i}_{q}^{ref}=\text{}\frac{2{T}_{ref}}{3p{\psi}_{m}},$

where:

*T*is the reference torque input._{ref}*p*is the number of pole pairs.*ψ*is the permanent magnet flux linkage._{m}

For operation below the base speed of the synchronous machine, ZDAC is a suitable
method. Above base speed, a field weakening controller is required to adjust the
*d*-axis reference.

To pregenerate optimal current references for several operating points offline, define two lookup tables using the user-defined lookup table approach:

${i}_{d}^{ref}=\text{f}\left({n}_{m},{T}_{ref},{v}_{dc}\right),$

and

${i}_{q}^{ref}=\text{g}\left({n}_{m},{T}_{ref},{v}_{dc}\right),$

where:

*n*is the rotor angular velocity._{m}*v*is the DC-link voltage of the converter._{dc}

To let the block create the lookup tables, choose the automatically generated lookup table approach. The block generates the lookup table using two strategies:

Maximum torque per ampere

Field weakening

The selection between the two strategies is based on the modulation index, which can be computed as follows:

$M=\frac{{V}_{s}}{k{V}_{ph\_max}},$

where *V _{s}* is the stator
voltage amplitude,

*k*is the modulation factor, and

*V*is the maximum allowable phase voltage. In the case that the modulation index is greater than 1, the block generates current references using the field weakening procedure. Otherwise, current references are computed using the maximum torque per ampere procedure.

_{ph_max}### Maximum Torque Per Ampere

You can generate current references in the constant torque region (occurring below rated speed) by using the maximum torque per ampere (MTPA) strategy.

The direct and quadrature components of the stator current are written in terms of angle and magnitude as:

${i}_{d}=-{I}_{s}\text{sin}\beta ,$

and

${i}_{q}=\text{}{I}_{s}\text{cos}\beta ,$

where:

*β*is the angle of the stator current vector.*I*is the stator current amplitude._{s}

Using the angle-magnitude variant of the d-q currents, the PMSM torque equation is written as:

${T}_{e}=\frac{3p}{2}{\psi}_{m}{I}_{s}\text{cos}\beta +\frac{3p}{4}\left({L}_{q}-{L}_{d}\right){I}_{s}{}^{2}\text{sin}2\beta ,$

where *L _{d}* and

*L*are the direct and quadrature inductances, respectively.

_{q}To obtain fast transient response and maximize torque with the smallest possible
stator current amplitude, MTPA imposes
(*dT _{e}*)/

*dβ*= 0 to the torque equation, which yields

$-\frac{3p}{2}{\psi}_{m}{I}_{s}\text{sin}\beta +\frac{3p}{2}\left({L}_{q}-{L}_{d}\right){I}_{s}^{2}\left({\mathrm{cos}}^{2}\beta -{\mathrm{sin}}^{2}\beta \right)=0.$

The MTPA *d*-axis current
*i _{d_mtpa}* is written in terms of the

*q*-axis component

*i*by substituting the d-q currents back from their angle and magnitude variants:

_{q_mtpa}${i}_{d\_mtpa}=\frac{{\psi}_{m}}{2\left({L}_{q}-{L}_{d}\right)}-\sqrt{\frac{{\psi}_{m}^{2}}{4{\left({L}_{q}-{L}_{d}\right)}^{2}}+{i}_{q\_mtpa}^{2}}.$

Finally, by plugging the previous equation into the d-q variant of the PMSM torque equation, the following polynomial is obtained:

$9{p}^{2}{\left({L}_{q}-{L}_{d}\right)}^{2}{i}_{q\_mtpa}^{4}+6{T}_{ref}p{\psi}_{m}{i}_{q\_mtpa}-4{T}_{ref}^{2}=0.$

The *q*-axis component is obtained by solving
this polynomial.

### Field Weakening

You can generate current references in the above rated speed region by using the field weakening (FW) strategy.

Above the rated speed, the stator voltage is limited by the power converter and the available DC-link voltage. The maximum stator voltage is:

${V}_{s}=\sqrt{{v}_{d}^{2}+{v}_{q}^{2}}\le {V}_{ph\_max},$

where *V _{ph_max}* is the
maximum available stator phase voltage.

The steady-state voltage equations for PMSMs are

${v}_{d}={R}_{s}{i}_{d}-{\omega}_{e}{L}_{q}{i}_{q}\text{},$

and

${v}_{q}={R}_{s}{i}_{q}+{\omega}_{e}\left({L}_{d}{i}_{d}+{\psi}_{m}\right).$

For rotor speeds above rated, the stator resistance is negligible, and the field
weakening *d*-axis current component
*i _{d_fw}* is obtained in terms of the

*q*-axis component

*i*from the

_{q_fw}*v*steady-state equation:

_{q}${i}_{d\_fw}=-\frac{{\psi}_{m}}{{L}_{d}}+\frac{1}{{L}_{d}}\sqrt{\frac{{V}_{ph\_max}^{2}}{{\omega}_{e}{}^{2}}-{\left({L}_{q}{i}_{q\_fw}\right)}^{2}},$

Finally, by plugging the *i _{d_fw}* equation
into the PMSM torque equation, the following polynomial is obtained:

$9{p}^{2}{\left({L}_{d}-{L}_{q}\right)}^{2}{L}_{q}^{2}{\omega}_{e}{}^{2}{i}_{q\_fw}^{4}+\left(9{p}^{2}{\psi}_{m}^{2}{L}_{q}^{2}{\omega}_{e}{}^{2}-9{p}^{2}{\left({L}_{d}-{L}_{q}\right)}^{2}{V}_{ph\_max}^{2}\right){i}_{q\_fw}^{2}-12{T}_{ref}p{\psi}_{m}{L}_{d}{L}_{q}{\omega}_{e}{}^{2}{i}_{{q}_{fw}}+4{T}_{ref}^{2}{L}_{d}^{2}{\omega}_{e}{}^{2}=0$

The *q*-axis component is obtained by solving this
polynomial.

### Assumptions

The machine parameters are constants.

### Limitations

The automatically generated current references introduce latency in the presimulation phase. For medium-power PMSM drives the latency is around 300 ms.

## Ports

### Input

### Output

## Parameters

## Model Examples

## References

[1] Haque, M. E., L. Zhong, and M. F. Rahman. "Improved trajectory control for an
interior permanent magnet synchronous motor drive with extended operating limit."
*Journal of Electrical & Electronics Engineering.* Vol. 22,
Number 1, 2003, p. 49.

[2] Carpiuc, S., C. Lazar, and D. I. Patrascu. "Optimal Torque Control of the
Externally Excited Synchronous Machine." *Control Engineering and Applied
Informatics.* Vol. 14, Number 2, 2012, pp. 80–88.

## Extended Capabilities

**Introduced in R2017b**